Maass forms and the mock theta function $f(q)$

Abstract

Let $f(q)=1+\sum_{n=1}^{\infty} \alpha(n)q^n$ be the well-known third order mock theta of Ramanujan. In 1964, George Andrews proved an asymptotic formula of the form $$\alpha(n)= \sum_{c\leq\sqrt{n}} \psi(n)+O_\epsilon\left(n^\epsilon\right),$$ where $\psi(n)$ is an expression involving generalized Kloosterman sums and the $I$-Bessel function. Andrews conjectured that the series converges to $\alpha(n)$ when extended to infinity, and that it does not converge absolutely. Bringmann and Ono proved the first of these conjectures. Here we obtain a power savings bound for the error in Andrews' formula, and we also prove the second of these conjectures. Our methods depend on the spectral theory of Maass forms of half-integral weight, and in particular on an average estimate which we derive for the Fourier coefficients of such forms which gives a power savings in the spectral parameter. As a further application of this result, we derive a formula which expresses $\alpha(n)$ with small error as a sum of exponential terms over imaginary quadratic points (this is similar in spirit to a recent result of Masri). We also obtain a bound for the size of the error term incurred by truncating Rademacher's analytic formula for the ordinary partition function which improves a result of the first author and Andersen when $24n-23$ is squarefree.